On pseudo-effective cones of projective bundles and volume function

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In this paper, we compute the pseudo-effective cones of various projective bundles [Formula: see text] over higher dimensional varieties [Formula: see text] under some assumptions on [Formula: see text] as well as on the vector bundle [Formula: see text]. We also compute the volume function on fiber product [Formula: see text] of two projective bundles over a smooth irreducible complex projective curve [Formula: see text]. In particular, we show that the volume function on the fiber product of two ruled surfaces is of polynomial type.